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Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Adaptive Control

Chapter 14: Adaptive regulation – Rejection of unknown disturbances

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Chapter 14: Adaptive regulation – Rejection of unknown disturbances

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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The Active Suspension System The Active Suspension System

controller

residual acceleration (force) primary acceleration / force (disturbance) 1 2 3 4 machine support elastomere cone inertia chamber piston main chamber hole motor actuator (piston position)

s Ts m 25 . 1 =

+ + −

A / B q-d ⋅ S / R D / C q

1

• d ⋅

u(t)

ce) (disturban (t) up

Controller

force) (residual y(t)

Plant

) ( p1 t

Two paths :

• Primary
• Secondary (double

differentiator) Objective:

• Reject the effect of unknown

and variable narrow band disturbances

• Do not use an aditional

measurement

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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The Active Suspension

Residual force (acceleration) measurement Active suspension Primary force (acceleration) (the shaker) Elastomer

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The Active Vibration Control with Inertial Actuator The Active Vibration Control with Inertial Actuator

s Ts m 25 . 1 =

+ + −

A / B q-d ⋅ S / R D / C q

1

• d ⋅

u(t)

ce) (disturban (t) up

Controller

force) (residual y(t)

Plant

) ( p1 t

Two paths :

• Primary
• Secondary (double

differentiator) Objective:

• Reject the effect of unknown

and variable narrow band disturbances

• Do not use an aditional

measurement

Same control approach but different technology

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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View of the active vibration control with inertial actuator

Residual force (acceleration) measurement Primary force (acceleration) (the shaker) Passive damper Inertial actuator (not visible)

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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View of a flexible controlled structure using inertial actuators

Inertial actuator

Inertial actuators Accelerometers

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Active Suspension

Primary path

Frequency Characteristics of the Identified Models

Secondary path

; 16 ; 14 = = = d n n

B A

Further details can be obtained from : http://iawww.epfl.ch/News/EJC_Benchmark/

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Direct Adaptive Control : disturbance rejection Closed loop Open loop

Initialization of the adaptive controller

Disturbance : Chirp

25 Hz 47 Hz

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Direct adaptive control

Time Domain Results Adaptive Operation

Simultaneous controller initialization and disturbance application

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Direct Adaptive Control

19.8 19.9 20 20.1 20.2 20.3 20.4 20.5 20.6

• 4
• 2

2 4 Time(s) Output (residual force) 19.8 19.9 20 20.1 20.2 20.3 20.4 20.5 20.6

• 2

2 Time(s) Input (control action) 32 Hz 20 Hz 29.8 29.9 30 30.1 30.2 30.3 30.4

• 2

2 Time(s) Output (residual force) 29.8 29.9 30 30.1 30.2 30.3 30.4

• 2
• 1

1 2 Time(s) Input (control action) 20 Hz 32 Hz

Output Input

Initialization of the adaptive controller Adaptation transient Adaptation transient

Step frequency changes

• utput

input

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Rejection of unknown finite band disturbances

• Assumption: Plant model almost constant and known (obtained by system

identification)

• Problem: Attenuation of unknown and/or variable stationary disturbances

without using an additional measurement

• Solution: Adaptive feedback control
• Estimate the model of the disturbance (indirect adaptive control)
• Use the internal model principle
• Use of the Youla parameterization (direct adaptive control)

A robust control design should be considered assuming that the model of the disturbance is known

Attention: The area was “dominated “ by adaptive signal processing solutions (Widrow’s adaptive noise cancellation) which require an additional transducer Remainder : Models of stationary sinusoidal disturbances have poles on the unit circle A class of applications: suppression of unknown vibration (active vibration control)

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Unknown disturbance rejection Unknown disturbance rejection – – classical solution classical solution

• requires the use of an additional transducer
• difficult choice of the location of the transducer
• adaptation of many parameters

+ +

A / B q-d ⋅

u(t)

Plant

) ( p1 t

) (t δ

p p D

N /

FIR

Disturbance Environment Disturbance correlated measurement Disturbance +Plant model Adaptation Algorithm

y(t)

Obj: minimization of E{y2}

Disadvantages: Not justified for the rejection of narrow band disturbances

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Notations

K G

v(t) p(t) y(t) u(t)

• v(t)

r(t)

) q ( A ) q ( B q ) q ( G

1 1 d 1 − − − −

= ) ( ) ( ' ) ( ) ( ' ) ( ) ( ) (

1 1 1 1 1 1 1 − − − − − − −

= = q H q S q H q R q S q R q K

S R

Output Sensitivity function :

) z ( R ) z ( B z ) z ( S ) z ( A ) z ( P

1 1 d 1 1 1 − − − − − −

+ =

Closed loop poles : ) ( ) ( ) ( ' ) ( ) (

1 1 1 1 1 − − − − −

= z P z H z S z A z S

S yp

The gain of Syp is zero at the frequencies where Syp(e-jω)=0 (perfect attenuation of a disturbance at this frequency)

+

yp

S

HR and HS are pre-specified

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Dirac circle; unit

• n the

poles e disturbanc tic determinis : ) ( ) ( ) ( ) (

1 1

= → ⋅ =

− −

δ(t) D t q D q N t p

p p p

δ

Deterministic framework Stochastic framework

) (0, sequence noise hite Gaussian w circle; unit

• n the

poles e disturbanc stochastic : ) ( ) ( ) ( ) (

1 1

σ = → ⋅ =

− −

e(t) D t e q D q N t p

p p p

Disturbance model

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). ( ) ( ' ) ( ); ( ) ( ' ) ( : Controller

1 1 1 1 1 1 − − − − − −

⋅ = ⋅ = q H q S q S q H q R q R

S R

Internal model principle: HS(z-1)=Dp(z-1) Closed loop system. Notations Closed loop system. Notations

) ( ) (q D 1 ) P(q ) (q ) (q )S' (q )H A(q y(t)

1

• p

1

• 1
• 1
• 1

S

• 1

t N p δ ⋅ ⋅ = + +

−

A / B q-d ⋅ S / R

u(t)

Controller Plant ) ( p t

) (t δ

p p D

N /

Dirac circle; unit

• n the

poles e disturbanc tic determinis : ) ( ) ( ) ( ) (

1 1

= → ⋅ =

− −

δ(t) D t q D q N t p

p p p

δ

) ( ) ( ) ( ) ( ) ( : poles CL ) ( ) ( ) ( ) ( ) ( ) ( ) ( : Output

1 1 1 1 1 1 1 1 1 − − − − − − − − − −

+ = ⋅ = ⋅ = q R q B z q S q A q P t p q S t p q P q S q A t y

d yp

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Two-steps methodology:

• 1. Estimation of the disturbance model,
• 2. Computation of the controller, imposing

) (

1 −

q Dp ) ( ˆ ) (

1 1 − −

= q D q H

p S

It can be time consuming (if the plant model B/A is of large dimension)

Indirect adaptive regulation Indirect adaptive regulation

Environment

+ +

A / B q-d ⋅

u(t)

Plant ) ( p t ) (t δ

p p D

N /

Disturbance Disturbance model

y(t)

Controller Disturbance Model Estimation Design Method Specs. Disturbance Observer

• )

( ˆ t p

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Indirect adaptive control Indirect adaptive control Step II: Computation of the controller

' ˆ ' ˆ ˆ S D S P BR q S D A D H

p d p p S

= = + =

−

Solving Bezout equation (for S’ and R)

Remark: It is time consuming for large dimension of the plant model

Step I : Estimation of the disturbance model ARMA identification (Recursive Extended Least Squares)

Adaptive Control – Landau, Lozano, M’Saad, Karimi

19 Plant Model Model

Central contr: [R0(q-1),S0(q-1)]. CL poles: P(q-1)=A(q-1)S0(q-1)+q-dB(q-1)R0(q-1). Control: S0(q-1) u(t) = -R0 (q-1) y(t) Q-parametrization : R(z1)=R0(q-1)+A(q-1)Q(q-1); S(q-1)=S0(z-1)-q-dB(q-1)Q(q-1).

Model = Plant w=Ap1

Q Q

n n q

q q q q q Q

− − −

+ + = .... ) (

1 1 1

Internal model principle and Q Internal model principle and Q-

• parametrization

parametrization) )

The closed loop poles remain unchanged Control: S0(q-1) u(t) = -R0 (q-1) y(t) - Q (q-1) w(t), where w(t) = A (q-1) y(t) - q-dB (q-1) u(t). CL poles: P(q-1)=A(q-1)S0(q-1)+q-dB(q-1)R0(q-1).

) ( ) ( ) ( ) (

1 1

t y q R t u q S

− −

− =

Adaptive Control – Landau, Lozano, M’Saad, Karimi

20 Plant Model

+ +

A B q d /

−

A B q d /

−

+

• A

Q R / 1 S

p p D

N /

) (

1 t

p ) (

1 t

p ) (t y ) (t u ) (t δ

Yula-Kucera parametrization An interpretation for the case A asympt. stable

) (t w

) ( ) ( t D AN t w

p p δ

=

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Internal model principle and Q- parameterization

Central contr: [R0(q-1),S0(q-1)]. CL Poles: P(q-1)=A(q-1)S0(q-1)+q-dB(q-1)R0(q-1). Control: S0(q-1) u(t) = -R0 (q-1) y(t) Q-parameterization : R(z-1)=R0(q-1)+A(q-1)Q(q-1); S(q-1)=S0(z-1)-q-dB(q-1)Q(q-1).

Closed Loop Poles remain unchanged

p d

MD BQ q S S = − =

−

S BQ q MD

d p

= +

−

Solve:

? ?

Internal model assignment on Q (find Q such that S contains the disturbance model):

Q can be used to “directly”tune the internal model without changing the closed loop poles(see next)

Will lead also to an « indirect adaptive control solution »

BUT:

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Goal: minimize y(t) (according to a certain criterion).

Direct Adaptive Control (unknown Dp)

Hypothesis: Identified (known) plant model (A,B,d).

[ ] [ ]

) ( ) P(q ) )Q(q (q q

• )

(q S ) ( ) (q D ) (q ) P(q ) )Q(q (q q

• )

(q S ) A(q y(t) e. disturbanc tic determinis : ) ( ) ( ) ( ) ( Consider

1

• 1
• 1
• d
• 1
• 1
• p

1

• 1
• 1
• 1
• d
• 1
• 1
• 1

1 1

t w B t N B t q D q N t p

p p p

= ⋅ ⋅ = ⋅ =

− −

δ δ Define (construct): S(q-1)

). ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

1 1 1 1 1

t w q Q q P q B q t w q P q S t

d

⋅ − ⋅ =

− − − − − −

ε ) ( ) ( t D AN t w

p p δ

=

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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S BQ q MD

d p

= +

−

We need to express ε(t)as:

[ ]

) ( ) , 1 ( ˆ ) ( ) 1 (

1 1

t q t Q q Q t Ψ + − = +

− −

ε

Using: , (*) becomes

) 1 ( ) ( ) ( ) ( )] , 1 ( ˆ ) ( [ ) 1 (

1 1 1 1

+ + ⋅ ⋅ + − = +

− − ∗ − − −

t v t w q P q B q q t Q q Q t

d

ε

Vanishing term

(*)

) 1 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) 1 (

1 1 1 1 1 1

+ = + = +

− − − − − −

t q P q N q M t w q P q D q M t v

p p

δ

) 1 ( ) ( ) ( ) ( ) ( ) (

1 1 1 1

+ ⋅ = ⋅

− − − − − ∗ −

t w q P q B q t w q P q B q

d d

Details:

Leads to a direct adaptive control

[ ]

Q i q t Q

t i Q

ˆ , min arg )* , ( ˆ

2 ˆ 1

∑ =

= −

ε

search recursively for : Instead of solving

S BQ q MD

d p

= +

−

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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: )) ( ) ( ) 1 ( ) ( ( ); ( ) ( ) 1 ( ) ( ) 1 (

1 1 1 1

define t u q B t u q B t u q B q t y q A t w

d − ∗ − − ∗ − −

= + − + = + ) ( ) 1 ( ˆ ) 1 ( ) 1 ( : error adaptation ) ( ) ( ˆ ) 1 ( ) 1 ( : error adaptation

1 1

t t t w t posteriori A t t t w t priori A

T T

φ θ ε φ θ ε + − + = + − + = +

Parameter adaptation algorithm:

⎪ ⎩ ⎪ ⎨ ⎧ + = + + + + = +

− −

). ( ) ( ) ( ) ( ) ( ) 1 ( 1); (t (t) 1) F(t (t) ˆ 1) (t ˆ

2 1 1 1

t t t t F t t F

T

φ φ λ λ ε φ θ θ

Various choices possible for λ1 and λ2 which define the adaptation gain time profile

(For a stability proof see Automatica, 2005, no.4 pp. 563-574)

The Algorithm

measured

). 1 ( ) ( ) ( ) , ( ˆ ) 1 ( ) ( ) ( ) 1 (

1 1 1 1 1

+ − + = + °

− − − − − −

t w q P q B q q t Q t w q P q S t

d

ε

[ ] [ ]

) 1 since 2 (for , ) 1 ( ) ( ) ( ; ) ( q ˆ ) ( q ˆ (t) ˆ ); ( ) ( ) ( ) ( ); 1 ( ) ( ) ( ) 1 (

2 2 1 1 1 2 1 1 1

• n

n n t w t w t t t t w q P q B q t w t w q P q S t w

P P

D Q D T T d

= = − = = ⋅ = + ⋅ = +

− − ∗ − − −

φ θ

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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^ Adaptation algorithm

Direct adaptive rejection of unknown disturbances

• The order of the Q polynomial depends upon the order of the disturbance model

denominator (DP) and not upon the complexity of the plant model

• Less parameters to estimate then for the identification of the disturbance model
• Operation in “self –tuning” mode (constant unknown disturbance)
• r “adaptive” mode (time varying unknown disturbance)

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Further experimental results on the active suspension Comparison between direct/indirect adaptive control

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Narrow band disturbances = variable frequency sinusoid ⇒ nQ = 1 Frequency range: 25 ÷ 47 Hz Nominal controller [R0(q-1),S0(q-1)]: nR0=14, nS0=16 Evaluation of the two algorithms in real-time Real Real-

• time

time results results – – Active Suspension (continuation) Active Suspension (continuation)

• The algorithm stops when it converges and the controller is applied.
• It restarts when the variance of the residual force is bigger than a given threshold.
• As long as the variance is not bigger than the threshold, the controller is constant.

Implementation protocol 1: Self-tuning Implementation protocol 2: Adaptive

• The adaptation algorithm is continuously operating
• The controller is updated at each sample

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Frequency domain results – direct adaptive method

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Frequency domain results – indirect adaptive method

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Self-tuning Mode Adaptive Mode Direct Adaptive Control

• Direct adaptive control in adaptive mode operation gives better

results than direct adaptive control in self-tuning mode

• Direct adaptive control leads to a much simpler implementation

and better performance than indirect adaptive control

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Active vibration control using an inertial actuator Real-time results

Rejection of two simultaneous sinusoidal disturbances

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Active vibration control using an inertial actuator

Primary path Secondary path

; 12 ; 10

Frequency Characteristics of the Identified Models

= = = d n n

B A

Complexity of secondary path:

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Frequency domain results – direct adaptive method

Rejection of two simultaneous sinusoidal disturbances

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Time Domain Results – Direct adaptive control Adaptive Operation

Simultaneous rejection of two time varying sinusoidal disturbances

Output

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Time Domain Results – Direct adaptive control Evolution of the Q parameters

Simultaneous rejection of two time varying sinusoidal disturbances

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Time Domain Results – Direct adaptive control Evolution of the control input

Simultaneous rejection of two time varying sinusoidal disturbances

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Comparison direct/indirect adaptive regulation

Time domain results – Adaptive regime

Active vibration control using an inertial actuator Indirect adaptive method Direct adaptive method

Direct adaptive control leads to a much simpler implementation and better performance than Indirect adaptive control

Adaptive Control – Landau, Lozano, M’Saad, Karimi

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Conclusions Conclusions

• Using internal model principle, adaptive feedback control

solutions can be provided for the rejection of unknown disturbances

• Both direct and indirect solutions can be provided
• Two modes of operation can be used : self-tuning and adaptive
• Direct adaptive control is the simplest to implement
• Direct adaptive control offers better performance
• The methodology has been extensively tested on:
• active suspension system
• active vibration control with an inertial actuator